# Expectation of supremum

Let $x(t)$ a real valued stochastic process and $T>0$ a constant. Is it true that:

$\mathbb{E}[\displaystyle{\sup_{t\in [0,T]}} |x(t)|] \leq T \displaystyle{\sup_{t\in [0,T]}}\mathbb{E}[|x(t)|]$ ?

-
Something smells fishy here. The LHS is scale-invariant, but the RHS is not, so we can rescale everything so that the RHS is arbitrarily small... –  Zhen Lin Sep 28 '11 at 11:02

Elaboration on the comment by Zhen, just consider $x(t) = 1$ a.s. for all $t$ and $T = 0.5$
If $x(t)$ is measured in inches and $T$ in seconds, then the left side would be in inches, whereas the right side would be in seconds times inches. So if we change the units of time from seconds to eons, the numerical value on the right gets smaller, but that on the left does not. The numerical value on the right can be made as close to $0$ as desired by making the units of time big enough, whereas that on the left remains fixed at a positive number.